System and method for conducting a game including a computer-controlled player

ABSTRACT

A system and method for conducting a game between at least one live player and at least one computer-controlled player includes executing a training program between at least two agents to generate probability weights correlating actions or meta-actions representing a set or sequenced set of actions with a probability that the action or meta-action will produce a game outcome meeting a specified criterion or specified criteria. A game is conducted in which at least one live player plays against at least one computer-controlled player in which the computer-controlled player selects actions at one or more of the decision nodes in the game based, at least in part, on the probability weights.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of U.S. patent application Ser. No.15/961,074, filed Apr. 24, 2018, entitled SYSTEM AND METHOD FORCONDUCTING A GAME INCLUDING A COMPUTER-CONTROLLED PLAYER, which is nowU.S. Pat. No. 10,621,830 issuing on Apr. 14, 2020, which is aContinuation of U.S. application Ser. No. 15/293,593, filed Oct. 14,2016, entitled SYSTEM AND METHOD FOR CONDUCTING A GAME INCLUDING ACOMPUTER-CONTROLLED PLAYER, which published as U.S. Patent PublicationNo. 2017-0032623 on Feb. 2, 2017, now U.S. Pat. No. 9,953,491 issued onApr. 24, 2018. U.S. application Ser. No. 15/293,593 is a continuation ofU.S. patent application Ser. No. 14/919,549, filed Oct. 21, 2015, asU.S. Publication No. 2016-0180655, entitled SYSTEM AND METHOD FORCONDUCTING A GAME INCLUDING A COMPUTER-CONTROLLED PLAYER. U.S. patentapplication Ser. No. 14/919,549 is a continuation of U.S. patentapplication Ser. No. 14/253,320, filed Apr. 15, 2014, published on Oct.23, 2014, as U.S. Publication No. 2014-0315625, entitled SYSTEM ANDMETHOD FOR CONDUCTING A GAME INCLUDING A COMPUTER-CONTROLLED PLAYER.Application Ser. No. 14/253,320 is a continuation of U.S. patentapplication Ser. No. 13/920,943, filed Jun. 18, 2013, published on Oct.24, 2013, as U.S. Publication No. 2013-0281195, now U.S. Pat. No.8,696,446, issued on Apr. 15, 2014, entitled SYSTEM AND METHOD FORCONDUCTING A GAME INCLUDING A COMPUTER-CONTROLLED PLAYER. ApplicationSer. No. 13/920,943 is a continuation of U.S. patent application Ser.No. 13/628,689, filed Sep. 27, 2012, published on Apr. 11, 2013, as U.S.Publication No. 2013-0090153, entitled SYSTEM AND METHOD FOR CONDUCTINGA GAME INCLUDING A COMPUTER-CONTROLLED PLAYER. Application Ser. No.13/628,689 is a continuation of U.S. patent application Ser. No.13/031,434, filed Feb. 21, 2011, published on Dec. 8, 2011, as U.S.Publication No. 2011-0300924, now U.S. Pat. No. 8,430,744, issued Apr.30, 2013, entitled SYSTEM AND METHOD FOR CONDUCTING A GAME INCLUDING ACOMPUTER-CONTROLLED PLAYER. U.S. patent application Ser. No. 13/031,434is a Continuation of U.S. patent application Ser. No. 11/810,827, filedJun. 7, 2007, entitled SYSTEM AND METHOD FOR CONDUCTING A GAME INCLUDINGA COMPUTER-CONTROLLED PLAYER, which issued as U.S. Pat. No. 7,892,080 onFeb. 22, 2011. Application Ser. No. 11/810,827 claims benefit of U.S.Provisional Application Ser. No. 60/862,628, filed Oct. 24, 2006,entitled PROGRAM FOR PLAYING TWO-PLAYER TEXAS HOLD'EM. U.S. Pat. Nos.10,621,830; 9,953,491; 8,696,446; 8,430,744; and 7,892,080, and PatentApplication Publication Nos. 2017-0032623; 2016-0180655; 2014-0315625;2013-0281195; 2013-0090153; 2011-0300924 are hereby incorporated byreference in their entirety.

TECHNICAL FIELD

The present invention relates to methods and systems for conducting agame. More particularly, the present invention is a system and methodfor conducting a game, such as the casino card game of poker, includingat least one computer-controlled player playing against at least onelive player.

BACKGROUND

The game of poker presents a serious challenge to artificialintelligence research. Uncertainty in the game stems from partialinformation, unknown opponents, and game dynamics dictated by a shuffleddeck. Add to this the large space of possible game situations in realpoker games such as Texas Hold'em, and the problem becomes verydifficult indeed. Human players, and even the best computer players, arecertainly not optimal in that idiosyncratic weaknesses associated withthe human or computer players can be exploited to obtain higher pay offsas compared to an approximating approach utilizing linear programmingtechniques.

Existing approaches to opponent modeling have employed a variety ofapproaches including reinforcement learning, neural nets and frequentstatistics. Additionally, earlier work on using Bayesian models forpoker has attempted to classify the opponent's hand into one of avariety of broad hand classes. They did not model uncertainty in theopponent's strategy, utilizing instead an explicit strategyrepresentation. This strategy was updated based on empirical frequenciesat play, but these models showed little improvement due to thisupdating. Other systems utilized Bayesian probabilistic models forHold'em poker games wherein the uncertainty in the game and the opponentwas completely modeled.

In certain circumstances and in certain games, it is desirable todecrease the element of skill element and increase the chance element.For example, when a game such as poker is implemented as a house-bankedgame, such as in a gaming machine, it may be desirable that the game bea game where chance, rather than skill, is a factor in determining theoutcome. That is, where a game is implemented in a gaming machine, theplayer competes against the gaming machine. If the gaming machine adaptsto the player's skill, then this arguably becomes a game of skill andmay not be desirable for two reasons. First, the gaming machine wouldarguably play “too well” and only expert players or players who areexceptionally fortunate would have a chance to win against the gamingmachine. Second, under some states' gaming regulations, games of skillare only allowed when the game is “player-banked” and only non-casino,human players are competing against one another. Thus, casino poker istypically played in poker rooms in which multiple players eachcontribute to a pot. The pot, minus a “rake” retained by the house forhosting the game, is awarded to the winning player.

Among the problems in modeling poker play are the large number ofunknowns, such as the cards that have not been dealt at any point in thegame, and the multiple options available to a player at various pointsin the game. Another unknown is the role of “bluffing” by a player.

For example, in a typical hand of Texas Hold'em poker, a single gameconsists of a number of stages separated by decision nodes where adecision must be made by a player. At a first stage, the pot is seeded.In one embodiment, a blind bet is placed by at least one player. Inalternate embodiments, multiple blind bets, e.g. a small blind and alarge blind, are received from different players. Optionally, the blindbet(s) rotates among players. In an alternate or additional optionalembodiment, each participating player may place an ante wager. The blindbet(s) and/or ante wagers are aggregated to a pot and each player isdealt a hand of cards.

In conventional Texas Hold'em, the hand consists of two cards. This istypically followed by a round of betting. In a typical game of TexasHold'em, the actions available to a player are to: (a) fold, e.g.terminate play by the player; (b) bet or check, if no other player haspreviously placed an additional wager, or call, if another player haspreviously placed an additional wager; or (c) raise, if another playerhas placed an additional wager and the player wishes to increase theamount of the additional wager. Bets, calls, and raises are aggregatedto the pot. In some versions, the number of raises, as well as the sizeof the bets and raises, is limited.

Additional stages are conducted in which community cards available toall the players on constructing the player's final hand are revealed. Inconventional Texas Hold'em, a total of five community cards are revealedin three stages, with each stage followed by a round of betting. Asknown in the art, three cards (referred to as the “flop”) are revealedin a stage, one card (referred to as the “turn”) is revealed in anensuing stage, and one card (referred to as the “river”) is revealed ina later ensuing stage. If players fold thereby leaving only one activeplayer, the active player wins. If more than one player remains in thegame through the revelation of all the community cards and the rounds ofbetting, each player forms a final hand using five of the seven cards(two in the player's hand plus five community cards) available to theplayer. The final hands are compared, and at least a portion of the potis awarded to the player with the highest ranking poker hand.

SUMMARY

The present invention includes systems and methods for conducting agame. According to an embodiment of the present invention, a system isprovided for conducting a game between a computer-controlled player andat least one live player. In one such optional embodiment, the gameincludes at least two decision nodes at which a decision is made. Thegame produces a game outcome such as, for example, a determination ofwhich of the computer-controlled player(s) and live player(s) won thegame. In an optional embodiment, the live player(s) andcomputer-controlled player(s) place one or more wagers and the gameoutcome is used to resolve the wagering.

The system includes a data processor. A data storage communicates withthe data processor. The data storage stores instructions executable bythe data processor, including a training program. The training programconducts the game among two or more agents controlled by the dataprocessor to select actions at the decision nodes to produce gameoutcomes. In a further optional embodiment, meta-actions are used inselecting actions at the decision nodes. In one optional embodiment,each meta-action represents a set of one or more actions. In a furtheroptional embodiment, each meta-action represents a sequence of two ormore actions.

The training program evaluates the game outcomes based on one or morepredetermined criteria. In one optional embodiment, the actions includeat least one wagering decision such that the game outcome includes a winor loss of wagers. In one such optional embodiment, the predeterminedcriteria for evaluating the game outcomes include a minimization of themaximum loss.

Based on the evaluation of the game outcomes produced in the gamesconducted between or among agents, the training program constructsprobability weights associating sets of one or more of the actions witha probability that the predetermined criteria will be met as a result ofthe set of one or more actions. In an optional embodiment, the trainingprogram may include a neural net adapted to increase the probabilitythat actions leading to a game outcome meeting the predeterminedcriteria will be selected and decrease the probability that actionsleading to a game outcome not meeting the predetermined criteria will beselected.

A system also includes a gaming device. The gaming device includes agaming device processor, a gaming device interface in communication withthe gaming device processor, and a gaming device data storage incommunication with the gaming device processor. The gaming device datastorage stores instructions executable by the gaming device processor toconduct the game. The instructions include a game program that receivesinput including a wager from the live player through the gaming deviceinterface. The game could take any form. For example, in an optionalembodiment, the game conducted by the training program and the gameprogram is a game in which at least one of the actions at one or more ofthe decision nodes includes a wagering action to place an additionalwager. Similarly, the training program and game program could conduct agame in which at least one of the actions at one or more of the decisionnodes includes a fold action to terminate conduct of the game.

In one optional embodiment, the gaming machine data storage stores theprobability weights; in another optional embodiment, the system includesa database storage device communicating with the gaming machine to storethe probability weights. The game program conducts the game to produce agame outcome by controlling the computer-controlled player against thelive player. The computer-controlled player selects at least one actionat one or more of the decision nodes based, at least in part, on theprobability weights. In an optional embodiment, the game programcontrols the computer-controlled player to select meta-actions in theform of a set of one or more actions, or a sequence of two or moreactions, at one or more of the decision nodes based, at least in part,on the probability weights. Optionally, the selection is a weightedrandom selection in which the selection of an action or meta-action israndom based on the probability weights. In an optional embodiment, eachdecision node in the game includes the same actions available to thecomputer-controlled player and the live player.

The game program evaluates the game outcome and resolves the wager. Forexample, in one optional embodiment, the game is a poker game in whichthe wager is contributed to a pot, the game outcome is determined bywhich of the computer-controlled player and live player wins the pokergame, and at least a portion of the pot is distributed to the liveplayer if the live player wins the poker game.

In an optional embodiment, the game program reconstructs the probabilityweights such that game outcomes produced by the gaming device alter theprobability weights. In another optional embodiment, the trainingprogram fixes the probability weights such that the game outcomesproduced by the gaming device leave the probability weights unaltered.

A method according to the present invention is directed to conducting agame between a computer-controlled player and at least one live player.As above, the game includes at least two decision nodes at which adecision is made to select one or more actions to produce a gameoutcome.

Probability weights are generated. In an optional embodiment, multipleiterations of the game are conducted among two or more agents controlledby a data processor. Each iteration uses a meta-action representing asequence of two or more of the actions at the decision nodes to producegame outcomes. The game outcomes are evaluated based on one or morepredetermined criteria. For example, in an optional embodiment, theactions include a wagering action such that the game outcome includes awin or loss of wagers, and the predetermined criteria include aminimization of the maximum loss. Optionally, the training programtrains a neural net by increasing the probability that actions leadingto a game outcome meeting the predetermined criteria will be selectedand decreasing the probability that actions leading to a game outcomenot meeting the predetermined criteria will be selected.

Meta-actions are associated with probability weights that thepredetermined criteria will be met as a result of the meta-actions. Inan optional embodiment, the probability weights are fixed such that thegame outcomes produced while conducting the game leave the probabilityweights unaltered.

The game is conducted. Input, including a wager, is received from thelive player. A game outcome is produced by controlling thecomputer-controlled player against the live player. In an optionalembodiment, a game is conducted in which each decision node includes thesame actions available to the computer-controlled player and the liveplayer. The computer-controlled player selects at least one action atone or more of the decision nodes randomly based, at least in part, onthe probability weights. In an optional embodiment, thecomputer-controlled player selects one or more meta-actions in the formof a set of one or more actions, or a sequence of two or more actions,based on the probability weights. The game actions could take any form.For example, in an optional embodiment, at least one of the actions atone or more of the decision nodes includes a wagering action to place anadditional wager. In another optional embodiment, at least one of theactions at one or more of the decision nodes includes a fold action toterminate conduct of the game.

The game outcome is evaluated and the wager is resolved. In an optionalembodiment, the probability weights are reconstructed such that gameoutcomes produced while conducting the game alter the probabilityweights.

For example, in an optional embodiment, the game is a poker game inwhich the wager is contributed to a pot. The game outcome is determinedby which of the computer-controlled player and live player wins thepoker game. At least a portion of the pot is distributed to the liveplayer if the live player wins the poker game. If thecomputer-controlled player wins the poker game, the computer-controlledplayer (such as through a gaming machine) retains at least a portion ofthe pot.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of a method according to an embodiment of thepresent invention;

FIG. 2 is a block diagram of a system according to an embodiment of thepresent invention;

FIG. 3 is a flow diagram of a training program and agents according toan embodiment of the present invention;

FIG. 4 is a flow diagram of a training program and agents according toan embodiment of the present invention;

FIG. 5 is a flow diagram of an optional embodiment of an agent accordingto an embodiment of the present invention;

FIG. 6 is a flow chart of a method for controlling an agent or acomputer-controlled player according to an embodiment of the presentinvention;

FIG. 7 is a front view of a gaming device according to an embodiment ofthe present invention;

FIG. 8 is a block diagram of a gaming device according to an embodimentof the present invention;

FIG. 9 is a flow diagram of a game program and computer-controlledplayer according to an embodiment of the present invention; and

FIG. 10 is a flow chart of a method for conducting a game according toan embodiment of the present invention.

DETAILED DESCRIPTION

Reference is now made to the figures wherein like parts are referred toby like numerals throughout. The present invention is directed to amethod and system for conducting a game between at least onecomputer-controlled player and at least one live player. Referring toFIG. 1, in its broadest terms, a method according to an embodiment ofthe present invention includes conducting 102 a game between agents(such as computer-controlled agents) to generate 104 probabilityweights. When the game is conducted 106 between a live player and acomputer-controlled player the computer-controlled player acts, at leastin part, based on the generated probability weights.

It is contemplated that any game could be conducted. For example, thegame could be any card game, tile game, dice game, or the like includingpoker and its variants, Pai Gow and its variants, blackjack, baccarat,or any other game or game type. Thus, the examples given below asrelated to a version of Texas Hold'em poker should be interpreted asillustrative rather than limiting. However, it is noted that any gamecould be used which includes at least one decision node at which acomputer-controlled player and a live player make a decision to selectan action to produce a game outcome. As may be appreciated, the actionsavailable (also referred to as candidate actions) at each decision nodemay change.

In an optional embodiment, the present invention may be applied to anembodiment of Texas Hold'em poker. The pot is seeded. In an optionalembodiment, this could occur by one or more players placing a blind bet.In such an optional embodiment, the blind bet or blind bets may rotateamong players. For example, in an optional embodiment, two blind bets, abig blind and a small blind may be received. In an alternate oradditional optional embodiment, the pot may be seeded by an ante wagerfrom one or more of the participating players.

As may be appreciated, any set of playing cards (including aconventional deck, truncated deck, or supplemented deck) could be used.In an optional embodiment, a single conventional poker deck of fifty-twocards is shuffled. Each player is dealt a hand of two cards. Afterdealing the hands to the players, a decision node is reached in whichone of the players must make a decision to select an action. In anoptional embodiment in which a big blind bet and small blind bet areused, player 2 (in this example, the player placing the big blind bet isreferred to as player 1 and the player placing the small blind bet isreferred to as player 2) has the option of folding (i.e. terminating hisparticipation in the game), calling (i.e. matching) the big blind bet,or raising the big blind bet. If player 2 chooses to call the big blindbet or raise the blind bet, the decision shifts to player 1. In oneoptional embodiment, player 1 may have the option to call or raise anyraise of the blind bet. In a further optional embodiment, player 1 mayhave the option to raise any call of the blind bet. For example, if theblind bet is two units, and player 2 calls the blind bet by wagering twounits, player 1 may have the option (in one optional embodiment) ofraising the call by wagering additional units. Thus, in an optionalembodiment, multiple actions may occur at a decision node for eachplayer. For example, if player 2 chooses to call the big blind bet, thenplayer 1 chooses to raise, player 2 must make an additional decision toselect a candidate action, e.g., call, raise, or fold, in response toplayer 1's raise. In an optional embodiment, the number of raises and/orthe size of the raises may be limited. In an alternate optionalembodiment, the bets and raises are not limited. In either case, thewagers are optionally accumulated to the pot.

This alternation continues until both of the players have wagered thesame amount (typically ended with a call) or one of the players folds.If a player folds, player is terminated as to that player. In anoptional embodiment including only two players, at least a portion ofthe pot is awarded to the player who did not fold. Conversely, if two ormore players remain in the game, community cards are revealed. Thecommunity cards may be revealed in a single stage or in multiple stages.In an optional embodiment in which the community cards are revealed in asingle stage, the revelation of the community cards may be followed by adecision node or the game outcome may be immediately evaluated and thewager resolved. In an optional embodiment in which the community cardsare revealed in multiple stages, a decision node may occur between someor all of the stages and, optionally, after the final stage.Alternatively, the stages may occur merely for entertainment valuewithout additional decision nodes.

The game outcome is evaluated. In an optional embodiment, the gameoutcome consists of a comparison of the highest ranking five-card pokerhand that may be formed by each player from that player's hand plus thecommunity cards. In an optional embodiment based on Texas Hold'em, theplayer may use any five of the seven cards available. In an optionalembodiment based on Omaha Hold'em, the player may be required to useboth the cards of the player's hand. It is contemplated that othervariations would also be possible.

The hands are compared according to a predefined ranking of hands. Thehand with the predefined ranking relationship to the other hand isdeclared the winning hand. For example, in an optional embodiment, astandard poker ranking of hands is used and the hand with the higherpoker ranking is declared the winning hand.

In an example of a game according to such an optional embodiment, afirst player places a big blind bet and a second player places a smallblind bet. It should be noted that in an optional embodiment, a blindbet may be received from fewer than all the participating players. Thefirst player receives a hand consisting of A

K♦ and a second player receives a hand of 5

4

. Since the first player placed the big blind bet, the second player hasa decision at the decision node to select one of three actions: call thebig blind bet, raise the big blind bet, or fold. In this example, thesecond player decides to raise the big blind bet by matching the amountof the big blind bet and adding an additional raise. The first playerhas a decision at the same decision node to select from one of threeactions: call the second player's raise, raise the second player'sraise, or fold. In this example, the first player decides to raise thesecond player's raise by matching the second player's raise and addingan additional raise. If the limit of raises had been met, i.e. becausethe limit of the raises is equal to two raises, the second player wouldhave the decision to select one of two actions: call the first player'sraise or fold. If the limit of raises has not been met, i.e. becausethere is no limit to the number of raises or the limit of the raises isgreater than two raises, play at the decision node shifts to the secondplayer, with the option to call the first player's raise, raise thefirst player's raise, or fold. In this example, the second playerdecides to call the first player's raise. The second player's decisionto call ends this decision node and the community cards are dealt. Asmentioned above, the community cards could be dealt in stages withadditional decision nodes interspersed between stages. In this example,five community cards consisting of A

K

2

7

7♦ are dealt in a single stage. Also, as noted above, a decision nodecould occur after the community cards are dealt. In this example, thedealing of the community cards is followed by the evaluation of the gameoutcome. The highest ranking five-card hand that the first player canform is A

A

K

K♦ 7

(with a poker ranking of two pair) and the highest ranking five-cardhand that the second player can form is A

K

5

4

2

(with a poker ranking of flush). In this example, conventional pokerhand rankings are used. Thus, the second player is awarded at least aportion of the pot because a flush outranks two pair.

Referring to FIG. 2, a system for implementing a game such as thatdescribed above includes a data processor 202 and a data storage 204storing instructions executable by the data processor 202. Moreparticularly, the instructions executable by the data processor 202include a training program. Referring generally to FIGS. 3-6, a trainingprogram conducts the game to be implemented using two or more agentsplaying against one another under the control of the data processor. Byconducting the game multiple times, probability weights are generated.

In an optional embodiment, agents are designed to act on the basis ofavailable information. That is, in such an optional embodiment, an agentbases its decision to select actions on its own hand and the currentdecision node. In game-theoretic terms this means that the agents act oninformation sets and represent behavioral strategies. The game isconducted multiple times to produce game outcomes and probabilityweights are generated to correlate actions to the probability that theaction will lead to a game outcome satisfying a predetermined criterionor criteria.

It is contemplated that the actions of the agents at the decision nodesmay be determined in a random fashion. That is, in an optionalembodiment, the actions of the agents at any given decision node areindependent of any prior trials or decision nodes.

In another optional embodiment, it is contemplated that the actions ofthe agents at the decision nodes may be determined in a weighted randomfashion based on prior trials. For example, in one optional embodiment,the agents may use a lagging anchor routine. If S represents the set ofinformation states that the agent may encounter and A(s) represents thefinite set of available actions at state s∈S, for each s∈S and a∈A(s),the agent would have a probability P(s, a) of applying action a atinformation state s. If the agent's behavior is parameterized by v∈V:P_(v) (s, a), where V is a closed convex subset of real numbers, theagent allows probability distributions over the set of legal actions fordifferent information states, and these probability distributions maydepend on a set of internal parameters of the agent (v). Thus, in suchan optional embodiment, the goal of the training program is to findparameter values v*∈V so that the probability distribution correspondingto the parameter values produces game outcomes satisfying apredetermined criterion or criteria. It should be noted that otherlearning routines, such as Q-learning, or the like, could be used tocontrol the agents, and thereby construct the probability table.

In an optional embodiment, the agent includes a neural net that takes asinput the game state and one or more candidate actions, and gives aprobability distribution of the candidate actions as an output. Whensuch an agent responds to a game state at a decision node, it evaluatesall available candidate actions and randomly chooses an action accordingto the probability distribution output from the neural net. For example,in an optional embodiment, the neural net may include a multi-layerperceptron design with one layer of hidden units and sigmoid activationfunctions. For updating, standard back-propagation of errors may be usedin the training thereof. In one such optional embodiment, the neural netmay include the following binary input units: thirteen units forrepresenting the card values, one unit for signaling identical suit ofthe cards, one unit for signaling a pair, eight units for signaling thesize of the pot, and three units signaling the candidate actions (fold,call, and raise). In an optional embodiment, described in greater detailbelow, the candidate actions may be combined into “meta-actions.” Inthis regard, “meta-actions” may, in one optional embodiment, representany set of one or more actions or, in another optional embodiment,represent a sequence of two or more actions. The output node of theneural net represents the probability weight that the agent assigns tothe action or meta-action. In an optional embodiment, the number ofhidden nodes may be set to twenty. In one such optional embodiment, theinternal parameters (v's) are the neural net output probabilities, whichmay be adjusted by the training program.

In an optional embodiment, the probability function for eachplayer/agent is a function of the neural net function for theplayer/agent. For example, in one such optional embodiment where theneural net function of a player/agent is denoted by B_(v)(s,a), theprobability function for that player/agent is

${P_{v}\left( {s,a} \right)} = {\frac{B_{v}\left( {s,a} \right)}{\sum\limits_{\overset{¯}{a} \in {A{(s)}}}{B_{v}\left( {s,\overset{¯}{a}} \right)}}.}$The same relationship could give the probability function for the otherplayer/agent(s).

As noted above, the game outcomes generated by the multiple iterationsare used to generate probability weights that associates actions at thedecision nodes to the probabilities that the actions will produce gameoutcomes that satisfy a predetermined criterion or criteria. In oneoptional embodiment, the predetermined criterion is the minimization ofmaximum loss (also referred to as a “minimax” criterion). That is, insuch an optional embodiment, the probability weights favor, i.e., givesa greater probability of occurrence to, actions or meta-actions thattend to minimize the maximum loss. In an optional embodiment, theprobability weights may include probability distributions for variousgame states, such that when a particular game state is encountered, aprobability distribution correlated to that game state is utilized.

This construction may occur manually. For example, the multipleiterations of the game may be conducted and the probabilities may betracked to produce the probability weights. For example, in the optionalembodiment shown in FIG. 3, a training program 402 generates probabilityweights 410 based on the conduct of a plurality of games by two agents,referred to in this example as “blue agent” 404 and “red agent” 406.Inputs to blue agent 404 include blue agent's cards 414 and the gamestate 412 (e.g. the betting state, any community cards, and the like)and inputs to red agent 406 include red agent's cards 416 and the gamestate 412. The training program conducts the game using the action(s) ormeta-action(s) 424, depending on the embodiment, selected by blue agentand the action(s) or meta-action(s) 426, depending on the embodiment,selected by red agent at the decision nodes encountered during the game.As previously suggested, the game outcomes in these games resulting fromthe action(s) or meta-action(s) taken by blue agent and red agent areevaluated against predetermined criterion or criteria by the trainingprogram 402 and the probability weights 410 are constructed to emphasizethose action(s) or meta-action(s) that were more likely to produce agame outcome meeting the predetermined criterion or criteria.

In another optional embodiment, the probability weights may be generatedby optimizing the agents to meet the predetermined criterion orcriteria, and using the probability output from the optimized agents.For example, in the optional embodiment of FIG. 4, this may take theform of the training program 402 including a learning program 418 thatupdates or adjusts the agents 404, 406 as the multiple iterations areconducted. In one such optional embodiment, agents 404, 406 play asample game to its conclusion. Then each agent 404, 406 performs a “whatif” analysis in which an additional game is completed by the agents 404,406 for each action not selected in the original game at each decisionnode visited. The outcomes of these hypothetical games provide estimatesof how successful alternative actions would have been. The agents thenmodify their behavior, optionally through a neural net, to reinforcethose actions that would have been the most successful, i.e. would havebeen more likely to produce a game outcome that satisfies thepredetermined criterion or criteria. In one such optional embodiment,this is accomplished through the use of training pattern feedback 420 tothe agents 404, 406 of input and desired output. If a given action ormeta-action 424, 426 appears more successful in producing a game outcomethat satisfies the predetermined criterion or criteria than the others,for the given game state, the agent 404, 406 is biased to apply it moreoften. This means that the training pattern feedback 420 could be givenby the training program's evaluation of the state-action pair offset bythe action's relative success compared to the other actions. Because ofthis relative nature of the feedback signals, there is a risk that theagent's outputs may drift toward zero or one, which hurts theback-propagation learning. In an optional embodiment, the agent'soutputs approximate probability distributions, and therefore adjustmentof the feedback signals in the training patterns is done accordingly.For example, in pseudo-code, an optional embodiment could includeinstructions as follows (where vector quantities are shown in boldface,keywords are displayed in boldface courier, and each agent's (calledBlue and Red) probability function is denoted by B(⋅, ⋅) and R(⋅, ⋅),respectively:

repeat Iteration times {

 play a game between agents called Blue and Red 

for

 each decision node n ε g 

 do { A ←

 legal actions at n 

  E ←

 outcomes of games resulting from actions A at n 

  if

 Blue on turn in n 

{ P ← B(s, A) } else { P ← R(s, A) } p_(sum) ← 1^(τ) P e ← P^(τ)E/p_(sum) E ← E − 1e F ← P + E − 1 (p_(sum) − 1) if

 Blue on turn in n 

{

 train B with patterns {(s, A),F}

 } else {

 train R with patterns {(s, A),F}

 } } }

Operations involving vectors are interpreted component-wise, so thenotation implies several for-loops. As an example, the statement

train B with patterns {(s, A), F}

may optionally be implemented as:

-   -   for (i=1 . . . length(A)) do {        train B with pattern ((s, Ai), Fi        }.

The vector E represents game outcomes for the agents in games thatexplore the different actions A in node n. In these games, the agents'hands and the community cards are held fixed with the actions changingin different iterations. E represents the outcome of the actual game asthe estimated outcome from taking the action chosen in that game. Thenumber e estimates the expected payoff for the player on turn, given hiscurrent probability distribution over the actions A. The statementE←E−1e normalizes E by deducting e from each component. F is thecalculated vector of feedback, and the term −1(p_(sum)−1) is included topush the probability function (B or R) towards valid probabilitydistributions.

The agents may optionally be evaluated to ensure that the actions ormeta-actions selected at each decision node lead to game outcomes thatsatisfy the predetermined criterion or criteria. For example, in oneoptional embodiment, where the predetermined criterion is play thatproduces game outcomes that minimize the maximum loss (also known as a“minimax” strategy), the agents may be measured using the performancemeasure of “equity against globally optimizing opponent” or Geq. The Geqmeasure is defined as the expected payoff when the agent plays againstits most effective opponent, e.g., the best response strategy ingame-theoretic terms. The Geq measure conforms with game theory in thesense that an agent applies a minimax strategy if, and only if, its Geqis equal to the game's value, which is the maximum Geq achievable.

In an optional embodiment, the agents are developed and conducted asseparate agents that compete against each other. In an optionalembodiment, the agents may be merged into a single agent that plays bothsides of any one game for the purpose of evaluation. That is, in onesuch an optional embodiment, a single game is implemented as a pair ofgames, with each agent playing both hands against the other agent, e.g.agent 1 plays hand A against agent 2 playing hand B, whilesimultaneously agent 1 plays hand B against agent 2 playing hand A. Inone optional embodiment, for the sake of variance reduction, the cardsmay be fixed in both games, so that both agents get to play the samedeal from both sides. The average of the two game outcomes is then takenas the merged game's outcome. In such an optional embodiment, theredefined pair of games has a value known to be zero, by symmetry (i.e.,in such an optional embodiment, the amount won must equal the amountlost in any game pair).

In an optional embodiment, reference players may be created forcomparison to evaluate an agent's performance. For example, in anoptional embodiment, a set of three reference players employingdifferent strategies (such as a balanced-player, an aggressive-player,and a random-player) may be created. In one such optional embodiment, abalanced-player is an estimate of a minimax-playing agent, anaggressive-player is an agent that rarely folds and raises often, and arandom-player is an agent that makes completely random actions, withuniform probabilities over actions. As noted in my article entitled “AReinforcement Learning Algorithm Applied to Simplified Two-Player TexasHold'em Poker” (incorporated herein by this reference), an agentoperating according to the procedure previously described appears toproduce probability weights that allows a computer-controlled player toplay in a manner that approximates a minimax strategy. It is noted thatthis reference should be construed as explanatory rather than limitingas the present invention contemplates any form of agent, whether or notutilizing a learning program while producing the probability weights.Moreover, even where a learning program is used, the present inventioncontemplates that the agent could conduct the games in any fashion, andmay be evaluated against any predetermined criteria, and should not belimited to the examples given herein.

As noted above, in controlling the agents, the training program maycontrol the agents to make each decision separately so that actions areselected singly. For example, in an optional embodiment in which adecision node includes one or more actions selected from call (orcheck), raise, or fold, the agent may select each action separately.Thus, in one such optional embodiment, an agent faced with a decision ata decision node in a particular game state may select an action toraise. If the responsive action from the opposing agent in that decisionnode is to raise, the agent would reevaluate and decide whether to call,raise, or fold. The separate treatment of each decision does notnecessarily preclude the use of prior actions or projected futureactions in selecting an action. For example, an agent may decide whetherto call, raise, or fold based on its prior action or on the actions itmay take (or its opponent may take) in the next decision or decisionnode. In an alternate optional embodiment, the decision at anyparticular game state may be made based on the game state without regardto prior actions and/or future projected actions.

In such an optional embodiment, the probability weights generatedcorrelate actions or groups of actions to a probability for selectionbased on whether the action or group of actions produced a game outcomemeeting the predetermined criterion or criteria. For example, at aparticular decision node at a particular game state, the three actions(A1, A2, and A3) may be available. The game outcomes that result in thegames in which those actions were taken would be evaluated and, in anoptional embodiment, normalized so that the probability distributionsadd to 1. If, for the sake of example, A1 has a probability distributionof 0.6, A2 has a probability distribution of 0.1, and A3 has aprobability distribution of 0.3. of producing a game outcome satisfyingthe predetermined criterion or criteria, probability weights such asthat shown in Table 1 could be generated:

TABLE 1 Action Probability of selection 1 0.6 2 0.1 3 0.3

Thus, in such an optional embodiment, the agent (if the probabilityweights are fed back into the agent) and/or the computer-controlledplayer (as discussed in greater detail below) would be more likely toselect A1 than A2 or A3 at a particular decision node. However, since A2and A3 are not precluded from selection, a certain amount of randomnessor “bluff” may be introduced into the control of the agent and/orcomputer-controlled player since those actions will, on occasion, beselected when a decision node in the same or similar game situation isreached.

In another optional embodiment, the training program may group multipleactions into “meta-actions.” Meta-actions could simply be sets of one ormore actions or, in a further optional embodiment, could be sequences oftwo or more actions. It should be noted that the set or sequence ofactions represented by a meta-action could occur at a single decisionnode or could take place across multiple decision nodes.

For example, in an optional embodiment in which a decision node mayinclude wager and fold actions, i.e. a player may have one or moreoptions selected from call (or check), raise, or fold, a meta-action forthat decision node could be the sequence “raise-raise-fold.” In oneoptional embodiment, this meta-action could represent the sequence ofactions at a single decision node which would mean that, at thatdecision node, the agent raises the bet, if the other agent does notcall or fold, the agent again raises the bet, and if the other agentdoes not call or fold, the agent folds. In another optional embodiment,this meta-action could represent the sequence of actions to be taken ateach point where a decision is called for, e.g. the agent raises thebet, then raises the bet again at the next point in the game (whether atthe same decision node or a later decision node) where a decision iscalled for, then folds at the next point in the game (again whether atthe same decision node or a later decision node) where a decision iscalled for.

In such an optional embodiment, the probability weights generated maycorrelate meta-actions to probabilities of selection (e.g.,probabilities that the meta-action will lead to a game outcomesatisfying the predetermined criterion or criteria). In an optionalembodiment, the probabilities are scaled such that, when they are alladded up, they equal 1. For example, in one optional embodiment, sixmeta-actions (MA1, MA2, . . . , MA6) may be defined. If MA3, forexample, had a probability distribution of 0.4, MA6 had a probabilitydistribution of 0.3 and the rest of the probability distributions weresignificantly less, with MA4 being approximately 0.1, MA2 beingapproximately 0.1, and MA5 being approximately 0.05, and MA1 beingapproximately 0.05, probability weights would be generated in which MA3would tend to be selected more frequently than any other meta-actionbecause it tended to produce a game outcome that satisfied thepredetermined criterion or criteria more frequently. In other words, inthis example, the probability weights generated by the agents wouldappear as shown in Table 2:

TABLE 2 Meta-action Probability of selection 1 0.05 2 0.10 3 0.40 4 0.105 0.05 6 0.30

It is noted, however, that in this optional embodiment, there may besome randomness or “bluff” incorporated because the remainingmeta-actions are not precluded from selection. In fact, in this example,MA3 is only slightly more likely to be selected than MA6. Thus, in suchan optional embodiment, the agent (if the probability weights are fedback into the agent) or the computer-controlled player (as discussed ingreater detail below) would not necessarily select the same meta-actionevery time a decision node in the same or similar game situation isreached.

For example, one such optional embodiment is shown in FIGS. 5 and 6. Itshould be noted that while the optional embodiment of FIGS. 5 and 6 aredescribed here as an agent, it is contemplated that acomputer-controlled player could also take the form shown in FIGS. 5 and6. Referring first to FIG. 5, an agent may include a plurality of neuralnets 802. Optionally, each of the neural nets 802 utilizes substantiallythe same architecture, but each has a unique and distinct set of neuralnet weights in an associated database 806. That is, in an optionalembodiment, each of the neural nets 802 is trained on a particular setof inputs for a particular set of outputs.

The game state, e.g., the state of the cards, the state of the bets, andthe like, are input on an input 810. In an optional embodiment, theagent includes a preprocessing routine performed by a preprocessor 812that basically determines nontrivial aspects. For example, thepreprocessing routine could examine at the status of a particular twocard hand received by the agent and calculate the “strength” of the rawhand, for example, by calculating the probabilities of all possiblehands that the opponent agent might have and assess a probability orstrength for that particular hand based on those two cards. It iscontemplated that such preprocessing may occur, or may be repeated, atvarious points in the game play, such as when additional community cardsare exposed. It should be noted that while FIG. 5 illustrates the use ofa preprocessor 812, it is contemplated that this need not be performedusing a separate operation, but could occur within a neural net 802, orother process.

Returning to the optional embodiment of FIG. 5, a vector of inputs(V_(I)), that includes the preprocessed states from the preprocessor 812and the game state 810, is output from a summing block 814. This inputvector 822 is communicated to the neural nets 802. In the optionalembodiment of FIG. 5, each of the neural nets 802 is state dependent.This state dependency means that each neural net 802 is concerned withonly certain game states. Stated another way, in an optional embodiment,each of the neural nets 802 is associated with a different portion ofthe input space, i.e., a different game state or set of game states. Forexample, in the very first step of the game, all of the community cardsare hidden and the input from those community cards is “0.” In anoptional embodiment, the neural net provided for such an initial gamestate may differ from a neural net provided for a later game state,e.g., when one or more community cards have been revealed.Alternatively, certain game states may not include a separate neuralnet, but share a neural net with other game states.

Optionally, the agent may be further simplified by considering thatthere are a limited number of actions that can be taken by the agent atany point and these are the available outputs from the neural net thatis associated with that portion of the game. Thus, in such an optionalembodiment, there is a restricted set of outputs for each of the neuralnets relative to the others. By training the neural net on a restrictedset of inputs and providing a restricted set of outputs, the agent canbe simplified.

In the optional embodiment of FIG. 5, for example, each of the neuralnets 802 is uniquely trained for a particular game state based input andrestricted set of outputs associated with the particular portion of thegame that it is dedicated to. The neural nets 802 output a vector output“V_(o)” that is input to a multiplexer 820. The selection input on themultiplexer 820 is received on input 822 and this is based upon thestate input vector V_(I).

Continuing with the example of FIG. 5, the multiplexer 820 is operableto select one of the neural nets 802 for the output thereof and onlythis neural net is actually run. That is, in this optional embodiment,there is no necessity to run the neural net engine for the other neuralnets, as they were not selected based on game state. As such, anoptional embodiment may include a single neural net engine with theneural net parameters selected based on which neural net 802 isselected. The selected neural net 802 is run and to produce an output ofcandidate actions or, in the case of FIG. 5, meta-actions 824.

As previously described, meta-actions are grouped actions such as thesequence “call-raise-fold.” This means that there may be, at a point inthe game, a decision by the agent to respond with a call or a checkwhich, if the opponent does not fold, then the agent will raise on thenext opportunity and, if the opponent does not fold, the agent willfold. This is a group of actions that are predetermined at a single timebased on probability distributions and termed as a “meta-action.”

The optional embodiment of FIG. 5 includes a stacking routine 826 thatallows for stacking of the candidate meta-actions. A random number 828is optionally utilized to select from among the available meta-actions824 for output of one meta-action 830. Broadly stated, in this optionalembodiment, the agent determines an input vector, selects a neural netbased on game state, and randomly selects one of the meta-actionstherefrom. By “randomly” it is contemplated that the selection may be aweighted-random selection where each of these meta-actions has aprobability distribution, such that certain meta-actions may be morelikely to be selected than others. In such an optional embodiment, arandom number generator is operable to select among these meta-actionsbased upon that probability distribution.

As previously discussed, agents may include a stacking routine 826 toassist in the selection of meta-action(s). In one such optionalembodiment, each meta-action represents a different series of actionsimplementing a different group of decisions and each meta-action has aprobability associated therewith. As discussed above, optionally, theprobabilities are scaled such that, when they are all added up, theyequal 1. In such an optional embodiment, a random number generatorgenerates a random number 828 between 0 and 1 and the meta-actioncorrelated to that random number 828 is selected from the stack.

Turning to FIG. 6, a flow chart illustrating an optional embodiment of amethod for selecting actions or meta-actions by an agent or acomputer-controlled player is illustrated. A method may start 1102 andproceeds to the assembly 1104 of a state vector. As discussed above, thestate vector optionally includes the game state. A neural net isselected 1106 based on the game state. Optionally, the game state ispreprocessed to extract 1108 non-trivial features and an input vectorV_(I) is created 1110. The input vector is communicated 1112 to theselected neural net. As noted above, in an optional embodiment, a singleneural net may be used to produce the same effect as multiple neuralnets by parameterizing a “generic” neural net to the selected neural netbefore operation. An output vector Vo of candidate meta-actions results1114 from the neural net. The meta-actions are stacked and scaled 1116according to the probabilities associated with each of the meta-actionsfor that neural net output. A meta-action is selected 1120 in aweighted-random fashion. The selected meta-action is executed 1118.

Turning to FIGS. 7 and 8, a system according to the present inventionalso includes a gaming device for conducting the game between a liveplayer and one or more computer-controlled players. The gaming deviceincludes a gaming device processor 604, a gaming device interface 602,and a gaming device data storage 608. In an optional embodiment, thegaming device may include a chassis 502 which has associated therewith agaming device interface 602. Since the gaming device could take anyform, including a gaming machine, personal computer, handheld device orpersonal digital assistant (“PDA”), cellular telephone, kiosk, terminal,or other form, the chassis 502 could be in any size, shape, or form. Thegaming device interface 602 receives input from a live player and mayincorporate any type of input device 504, including a touchscreen,button panel, pointer, mouse, keyboard or key pad, or any other form.Optionally, the gaming device interface 602 also includes a display 510.In one optional embodiment, the display 510 and input device 504 areseparate from one another; in an alternate optional embodiment, thegaming device interface 602 may be a touchscreen or other device inwhich the input device 504 and display 510 are integrated. Optionally,the gaming device may further include a wager receiver 506 that receiveswagers in the form of coin, currency, tickets, vouchers,machine-readable cards, or the like. The gaming device may optionallyinclude an award handler 508 issue awards in the form of coin, currency,tickets, vouchers, machine-readable cards, or the like, that are earnedby a live player as a result of a game outcome.

As may be appreciated, a display 510 may display indicia used in thegame (such as cards, tiles, dice, or other game indicia) along withwagers and other information. For example, an optional embodimentdirected to poker or other forms of card games, the display 510 mayinclude a display of the cards in the hand dealt to the player. Asdiscussed above, for example, in a Texas Hold'em a player hand of twocards will be displayed to the live player. In an optional embodiment,the computer-controlled player or player(s) will not have “knowledge” ofthe live player's cards. Optionally, the display 510 may also include arepresentation of each computer-controlled player's cards displayed asif they were face-down or otherwise concealed from the live player'sview. In an optional embodiment directed to a game in which communitycards are used, the display 510 may include a representation of anycommunity cards face-down or otherwise concealed from the live player'sview, and may include animation to reveal the cards such as turning theconcealed cards over.

As previously noted, a gaming device includes a gaming device processorand a gaming device data storage. However, it should be noted that it isnot necessary that each gaming device have a separate and independentgaming device processor and/or gaming device data storage and it iscontemplated that, in an optional embodiment, a plurality of gamingdevices could share a gaming device processor and/or a gaming machinedata storage. Similarly, it is noted that the gaming device processorand/or gaming device data storage need not be local to the gaming devicebut could be remote from the gaming device and communicate with thegaming device via a wire or wireless communication link, networkconnection, Internet connection, or the like.

With continued reference to FIGS. 7 and 8, a gaming device according toan embodiment of the present invention may include a display 510 andinput device 504 interfaced through a gaming device interface 602 which,in turn, communicates with a gaming device processor 604. The gamingdevice processor 604 communicates with a gaming device data storage 608storing instructions executable by the gaming device processor 604 toconduct a game. Specifically, the instructions include a game programthat controls one or more computer-controlled players against one ormore live players who play a game via the gaming device interface 602.

Referring generally to FIG. 9, at least one of the computer-controlledplayer(s) conducts the game by making at least a portion of thedecisions at the decision nodes based on the probability weightsgenerated by the training program. In an optional embodiment, thecomputer-controlled player(s) may take actions singly or may makeactions in sets or sequenced sets, e.g. as meta-actions.

For example, in an optional embodiment, the computer-controlledplayer(s) are controlled (optionally through a neural net) so that thoseactions more likely to lead to a game outcome meeting the predeterminedcriterion or criteria according to the probability weights are morelikely to be selected. In one such optional embodiment, probabilityweights define the mapping of information that is received on the inputof a neural net at a computer-controlled player to the output. In suchan optional embodiment, the output of these neural nets is a probabilitydistribution for certain actions. That is, there are a number ofpossible actions associated with each decision to be taken at a decisionnode, e.g. fold, call, raise, and the like, with each decision nodeincluding one or more actions, e.g. fold, call, or raise on the firstdecision, then fold, call, or raise if the opponent raises, and thelike. Each of these actions at each of these actions is associated witha probability distribution. In such an optional embodiment, the actionsmay be made separately, in series, such that each decision is evaluatedaccording to the probability weights and the inputs (such as the gameconditions at the particular decision node). In an optional embodiment,a computer-controlled player may be controlled to make a decision bymaking a weighted-random selection based on the probability weightsgenerated.

In another example, actions may be grouped into sets equivalent oranalogous to the meta-actions described above with respect to anoptional embodiment of the training program. That is, in such anexample, the computer-controlled player(s) may be controlled bymeta-actions, i.e., a set of one or more actions (optionally a sequencedset of two or more actions), determined according to the inputs, such asthe conditions existing at the decision node, and the probabilityweights so that the meta-actions more likely to lead to a game outcomeconsistent with the predetermined criterion or criteria.

In an optional embodiment, the computer-controlled player may becontrolled to select a meta-action randomly based on the probabilityweights generated. As with the training program, the meta-actions couldcomprise a set of actions to be taken at a particular decision node orcould be a set of actions that could occur across two or more decisionnodes.

For example, in an optional embodiment, when a computer-controlledplayer receives a hand of K♥ J♥, probability weights may indicate thatfive meta-actions (MA1, MA2, . . . , MA5), are associated with thefollowing probabilities for leading to a game outcome consistent withthe predetermined criterion or criteria: MA1: 0.05, MA2: 0.1, MA3: 0.5,MA4: 0.05, MA5: 0.3. In this situation, a random selection is made fromamong the five meta-actions based on the probability weights. Thus, inthis case, MA3 would be the most likely to be selected, although MA5,MA2, or MA1 or MA4 could also be selected, in order of decreasinglikelihood.

Such a random selection could be accomplished in many different ways. Inone optional embodiment, for example, the game program may include, ormay communicate with, a random number generator. The random numbers thatmay be selected may be allocated among the possible meta-actions basedon the probability weights. Thus, in the example given, 5% of the randomnumbers may be allocated to MA1, 10% of the random numbers may beallocated to MA2, 50% of the random numbers may be allocated to MA3, 5%of the random numbers may be allocated to MA4, and 30% of the randomnumbers may be allocated to MA5. A random number is generated, and themeta-action to which the random number is allocated is selected for theconduct of the computer-controlled player. It is noted in an optionalembodiment having multiple computer-controlled players, thecomputer-controlled players may be conducted using the same probabilityweights or different probability weights; using the same random numberallocation or different random number allocations; using the same randomnumber or different random numbers; or may otherwise be conductedseparately from, or interdependently with, one another.

The probability weights could be stored locally, e.g. at the gamingdevice, or may be stored in a database storage device (not shown)separate from the gaming device. For example, the database storagedevice could be a file server, network server, remote storage, or thelike that is separate from, and in communication with, the gamingdevice.

In an optional embodiment, the probability weights may be fixed by thetraining program. In such an optional embodiment, the game program isunable to alter the probability weights through any “experience” gainedthrough conducting the computer-controlled player(s) against the liveplayer in the game. In such an optional embodiment, the probabilityweights followed by the computer-controlled player may be “fixed” suchthat for a given set of inputs, the probability distribution will alwaysbe the same.

In an alternate optional embodiment, the probability weights may beupdated by the game program based on the game outcomes occurring whenthe game is conducted between the computer-controlled player(s) and thelive player. Such updating could occur at any time in the conduct of thegame, including on a periodic basis, a real-time basis, or any otherbasis.

Referring to the conduct of the game, the game may take any form and mayuse any type of game indicia. While the example given below refers to acard game and the use of playing cards, it is noted that this isillustrative only and should not be interpreted as limiting.

In an optional embodiment, the game is a symmetric game. As may beappreciated, “symmetric” could have many different meanings. Forexample, as shown in FIG. 6, the game may be a game where the payoffsfor playing a particular strategy depend only on the strategiesemployed, not on who is playing them. That is, if one can change theidentities of the players without changing the payoffs or thestrategies, then the game may be termed “symmetric.” As a subset of sucha “symmetric” game, the games may include games in which the sameactions are available to each of the live player and thecomputer-controlled player at any decision node. It is also noted that a“symmetric” game may include games in which the symmetry occurs over thelong term, e.g. there are differences in the situation each player findshimself or herself in, but these even out over the long run. Forexample, a game including a rotating blind bet on each hand would, inany single game, have different strategies for the player with the blindbet and the player without the blind bet. However, over the long run,each will have the blind bet roughly the same number of times.

A game is “zero-sum” if the win of one side is equal to the loss of theother. As such, two-player Texas Hold'em with a limit on the wageringcould be characterized as a symmetric, zero-sum game. Where a blind betis included, the game program may take the form of a mixed strategy.This implies that, if the live player applies the same mixed strategy asthe one represented by the game program, i.e., using the probabilityweights, that live player would in the long term break even. Thistheoretically occurs because the blind bet rotates between the liveplayer and the computer-controlled player(s). Therefore, any advantagethat may be had shifts as the blind bet shifts. Thus, an optionalembodiment of the present system may be characterized an electronicgaming device that allows a live player to play a symmetric zero-sumgame against a fixed mixed strategy, where payoffs are given andmonetary units handled by the device. In such an optional embodiment,the probability that an action will be taken in response to a decisionis distributed over several actions. Such an optional embodimentprovides a situation, from the perspective of the live player, where thelive player is perceived to be on equal footing as thecomputer-controlled player.

The game program optionally includes a program module for handling thegame indicia, such as playing cards. For example, in an optionalembodiment, the game program includes a program module to randomize anddeal cards (or electronic representations thereof) to one or more liveplayers and one or more computer-controlled players. Thus, in oneoptional embodiment, the game program includes a program module toshuffle cards and select from the shuffled deck a defined quantity ofcards for the game. In an example directed to two-player Texas Hold'em,for example, nine cards would be selected, e.g. dealt, from the deck—twofor the live player's hand, two for the computer-controlled player'shand, and five for the community cards.

The game program also tracks the wagers and rewards. For example, in anexample directed to Texas Hold'em, the game program may include aprogram module to track the pot value by tracking the wagers (callwagers, blind bets, raises, and the like) and aggregating them in a pot.Upon certain game outcomes, at least a portion of the pot may be awardedto the player such as through a game credit register, through a rewardhandling device, or the like. It is noted that in an optional embodimentof the computer-controlled player, the game program would be wageringhouse funds and, thus, the game could be house-banked. The live player,on the other hand, could be required to cover any wagers (blind bets,call wagers, raises, and the like) by maintaining a positive balance ina credit register at the gaming device, depositing money, or accessing acredit or wagering account sufficient to cover any wagers. In anoptional embodiment, the game program may maintain a running balance andsettle with a live player at the end of a play session. In such anoptional embodiment, the player's balance cash, voucher, coded card, orthe like may be issued to the live player when the live playerterminates play.

Referring to the optional embodiment of a gaming device of FIG. 7, adisplay for a game of two-player Texas Hold'em may include communitycards 702 initially displayed face-down, a computer-controlled playerhand 710 that is known to the game program but is displayed face-down tothe live player, and a live player hand 712 that is visible to the liveplayer, but is unknown to the computer-controlled player, i.e. in suchan optional embodiment the live player hand is not provided as an inputto the computer-controlled player in making its actions.

Referring to FIGS. 7-9, a game program 902 is shown separate from acomputer-controlled player 906. It should be appreciated that the gameprogram 902 and the program controlling the computer-controlled player906 may be executed by a single gaming device processor 604. In anoptional embodiment, however, certain information provided to the gameprogram 902 is input to the computer-controlled player 906.

For example, in the optional embodiment of FIGS. 7-9, a game program 902may communicate data representing live player cards 914 and informationrepresenting the game state 912 (such as pot size, blind bet size, callbet size, raise bets, and the like) to a display controller 904controlling a display 510. The game program 902 may receive live playeraction(s) 924, such as through a gaming device interface 602 controllingan input device 504. The computer-controlled player 906 may receiveinputs in the form of the computer-controlled player's cards 916 and thegame state 912. In an optional embodiment including community cards 920,data representing the community cards 920 may be displayed at thedisplay controller 904 and provided as an input to thecomputer-controlled player 906. The computer-controlled player 906 hasaccess to the probability weights 910 and selects one or more actions(or meta-actions, depending on the optional embodiment) at one or moredecision nodes based at least in part on the probability weights 910 forthe particular game state, e.g. the state formed by one or more of thecomputer-controlled player cards 916, the game state 912 (including anylive player actions 924), and any community cards 920. For example, inan optional embodiment, the computer controlled player 906 makes aweighted random selection of an action where the weights are extractedfrom the probability weights 910 at the game state. Selectedcomputer-controlled player actions 926 may be provided to the gameprogram 902. Conduct of the game by the game program 902, based on thelive player actions 924, computer-controlled player actions 926, andcards dealt, produces a game outcome 908.

For example, in an optional embodiment shown in FIG. 10 directed toTexas Hold'em, the pot is seeded, in this example, through the receipt1002 of at least one blind bet. As noted above, the blind bet (or blindbets) optionally rotates among the players (both live andcomputer-controlled) so that over the long run, the game issubstantially symmetrical. Cards are dealt 1004 to the live player(s)and computer-controlled player(s). In one optional embodiment, two cardsare dealt to each player (both live and computer-controlled). In thisoptional embodiment, a decision node is reached. If, at the decisionnode, it is a live player's turn 1006, an action is received 1008 from alive player; if, at the decision node, it is not a live player's turn1006, an action is randomly selected 1010 for the computer-controlledplayer based, at least in part, on the probability weights generated.

In an optional embodiment, the actions may include wagering actions,such as bet, check, call, or raise. Optionally, wagers received throughsuch wagering actions are received into a pot. It is noted that withthese actions, the game state may change, thereby changing the gamestate input to a computer controlled player.

Additionally, in an optional embodiment, the actions may include a folddecision where the player (both live and computer-controlled) mayterminate participation in the game. If all but one player folds, thegame is terminated 1012 the wagers, e.g. the pot, are resolved. Forexample, in one optional embodiment, at least a portion of the pot isawarded to the one player (whether live or computer-controlled) who didnot fold.

If the game has not terminated 1012, a determination is made whether thedecision node is complete 1014. That is, in an optional embodiment, suchas where the decision node constitutes a round of wagering, multipleactions may occur in a single decision node. For example, if a player(live or computer-controlled) chooses to raise a wager, the otherplayers may have the option to call, raise, or fold in response to theraise. If the decision node is not complete 1014, additional actions arereceived from the players until the decision node is complete 1014.

In an optional embodiment directed to Texas Hold'em, community cards arerevealed or dealt 1018 after a first round of wagering. As previouslydiscussed, a simplified version may include revealing or dealing allcommunity cards together, and completing the game without 1020 anyadditional decision nodes. In an alternate optional embodiment, thecommunity cards are revealed or dealt in stages, with each stageseparated by a decision node. In such an optional embodiment, adetermination 1016 is made whether cards are to be revealed or dealt. Ifso, the cards are revealed or dealt 1018. For example, in an optionalembodiment of Texas Hold'em, an initial round of wagering may befollowed by the “flop,” that is, the revealing or dealing of threecommunity cards. After the cards are dealt or revealed, a determination1020 is made whether an additional decision node occurs. In an optionalembodiment of Texas Hold'em, the flop is followed by a decision nodethat is conducted similarly to the initial round of wagering. After thisround of wagering, the “turn” occurs, that is, one additional communitycard is dealt or revealed, followed by another round of wagering. Thisround of wagering is followed by the “river”, that is, one finalcommunity card is dealt or revealed, followed by another round ofwagering. At this point, no further cards remain to be dealt orrevealed, and no further decision nodes remain. The wagers are resolved,optionally by awarding 1022 at least a portion of the pot to the playerwith the highest ranking poker hand.

While certain embodiments of the present invention have been shown anddescribed it is to be understood that the present invention is subjectto many modifications and changes without departing from the spirit andscope of the invention presented herein.

What is claimed is:
 1. A method for use by a computer system for conducting a game between at least one computer-controlled player and at least one live player comprising: receiving, by the computer system, a plurality of predictions from a neural network, wherein the predictions represent a likelihood that each of a plurality of available actions will satisfy a predetermined criterion; selecting, by the computer system, one of the available actions for the computer-controlled player, wherein each of the available actions has a percentage chance of being selected that is equal to the likelihood of that available action satisfying the predetermined criterion; and evaluating, by the computer system, an outcome of a decision node in the game based on one of the available actions selected by a live player and the available action randomly selected for the computer-controlled player.
 2. The method of claim 1, wherein the predetermined criterion is a minimization of maximum loss criterion.
 3. The method of claim 1, wherein the likelihood of each action being selected is based on a probability weight generated by a neural network training program.
 4. The method of claim 1, further comprising: identifying, by the computer system, that the decision node in the game has been reached; and providing, by the computer system, a plurality of inputs to the neural network, wherein the inputs include a face value and suit of each of a plurality of community cards, a face value and suit of at least one computer-controlled card, and the plurality of available actions.
 5. The method of claim 4, further comprising: identifying, by the computer system, a game state of a plurality of game states at the decision node; and using, by the computer system, a predetermined probability distribution of probability weights corresponding to the identified game state for the available actions.
 6. The method of claim 5, wherein wagers are made at decision nodes during the game, and wherein the wagers are contributed to a pot.
 7. The method of claim 6, further comprising determining the outcome and distributing at least a portion of the pot to the live player.
 8. The method of claim 4, wherein at least one of the available actions is a meta-action that represents a sequence of two or more actions.
 9. The method of claim 8, wherein only a first portion of the sequence of two or more actions is executed at the decision node.
 10. The method of claim 9, wherein a second portion of the sequence is executed at another decision node.
 11. The method of claim 10, wherein execution of the second portion of the sequence at another decision node is conditional.
 12. A system for conducting a game between at least one computer-controlled player and at least one live player comprising: a data processor; a data storage; and a plurality of instructions stored in the data storage and executable by the data processor, the instructions including: instructions for receiving, by the system, from a neural network, a likelihood that each of a plurality of available actions will satisfy a predetermined criterion based on a game state; and instructions for selecting one of the available actions for the computer-controlled player, wherein each of the available actions has a likelihood of being selected that is equal to the likelihood of that available action satisfying the predetermined criterion.
 13. The system of claim 12, wherein the predetermined criterion is a minimization of maximum loss criterion.
 14. The system of claim 13, further comprising instructions for receiving wagers at decision nodes during the game, and instructions for adding the wagers are to a pot.
 15. The system of claim 14, further comprising instructions for determining a game outcome and instructions for distributing at least a portion of the pot to a live player.
 16. The system of claim 12, further comprising: instructions for identifying that a decision node in the game has been reached; and instructions for providing the game state at the decision node and the plurality of available actions to the neural network.
 17. The system of claim 16, further comprising: instructions for identifying a game state of a plurality of game states at the decision node; and instructions for using a predetermined probability distribution of probability weights corresponding to the identified game state for the available actions.
 18. The system of claim 16, wherein at least one of the available actions is a meta-action that represents a sequence of two or more actions.
 19. The system of claim 18, wherein only a first portion of the sequence is executed at the decision node.
 20. The system of claim 19, wherein a second portion of the sequence is executed at another decision node. 